Deformation of Dirac operators along orbits and quantization of noncompact Hamiltonian torus manifolds

Abstract We give a formulation of deformation Dirac operator along orbits group action on possibly noncompact manifold to get an equivariant index and K-homology cycle representing the index. apply this framework Hamiltonian torus manifolds define geometric quantization from viewpoint theory. two applications. The first one is proof [Q,R]=0 type theorem, which can be regarded as Vergne conjecture for abelian case. other Danilov-type formula toric case in setting, localization phenomenon setting. proofs are based lattice points.